n is an integer from 1 to 50, what is the probability that n(n+1) is (a) a multiple of 5? (b) a multiple of 6?

Soln:

For the number n(n+1) to be divisible by 6 means it should be divisible by 2 and 3 both.

n(n+1) is always divisible by 2 if n is integer.

Now, n(n+1) should be divisible by 3 for that either n should be multiple of 3 or n+1 should be multiple of 3.

So, till 50 the numbers such that n will be multiple of 3 are 3 6 9.........48 = 16 numbers

Similarly, n+1 will be divisible by 3 if n = 2 5 8 ...............50 = 17 numbers, there are no numbers in common.

So, total favourable items will be 16 + 17 = 33 and total possible items are 50 hence probability is 33/50.